On the randomized Euler schemes for ODEs under inexact information

Abstract

We analyse errors of randomized explicit and implicit Euler schemes for approximate solving of ordinary differential equations (ODEs). We consider classes of ODEs for which the right-hand side functions satisfy Lipschitz condition globally or only locally. Moreover, we assume that only inexact discrete information, corrupted by some noise, about the right-hand side function is available. Optimality and stability of explicit and implicit randomized Euler algorithms are also investigated. Finally, we report the results of numerical experiments which support our theoretical conclusions.

Appendix

The following lemma can be proven in the same fashion as Lemma 1(i) in [1].

Lemma 2

Let $(\eta ,f)\in F^{{\varrho }}_{R_{2}}$, where

$$ R_{2} = K(1+b-a)e^{K(b-a)} + K. $$

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Then

• (1) has a unique solution z = z(η, f) such that $z\in \mathcal {C}^{1}\left ([a,b]\times \mathbb {R}^{d}\right )$ and z(t) ∈ B(η, R₂) for all t ∈ [a, b];

• there exist $C_{1} = C_{1}(a,b,K)\in (0,\infty )$ and $C_{2} = C_{2}(a,b,K,L)\in (0,\infty )$ such that for all t ∈ [a, b]

  $$ \|z(t)-z(s)\| \leq C_{1} |t-s|, $$

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  $$ \left\|z^{\prime}(t)-z^{\prime}(s)\right\| \leq C_{2}|t-s|^{{\varrho}}. $$

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Next lemma is used to show existence and uniqueness of a measurable solution to the implicit randomized Euler scheme. Its proof can be found in [3] (Lemma 4.3).

Lemma 3

Let $\tilde {\mathcal {F}}$ be a complete sub σ-algebra of the σ-algebra Σ, $M \in \tilde {\mathcal {F}}$ with $\mathbb {P}\left (M\right ) = 1$ and $h\colon {\varOmega }\times \mathbb {R}^{d} \to \mathbb {R}^{d}$ such that the following conditions are fulfilled.

• The mapping x↦h(ω, x) is continuous for every ω ∈ M.

• The mapping ω↦h(ω, x) is $\tilde {\mathcal {F}}$-measurable for every $x\in \mathbb {R}^{d}$.

• For every ω ∈ M there exists a unique root of the function h(ω,⋅).

Define the mapping

$$Q\colon {\varOmega} \ni \omega \mapsto Q(\omega) \in \mathbb{R}^{d},$$

where Q(ω) is the unique root of h(ω,⋅) for ω ∈ M and Q(ω) is arbitrary for ω ∈Ω∖ M.

Then Q is $\tilde {\mathcal {F}}$-measurable.